Heat olive oil in a large nonstick skillet over medium heat. Add mushrooms and onion; cover and cook 7 minutes or until onion is tender, stirring occasionally. Remove from heat; cool.
Combine the mayonnaise, basil, and black pepper. Spread 1 tablespoon mayonnaise mixture on each bread slice; layer each of 2 slices with 1/3 cup bell peppers, 1/2 cup mushroom mixture, and 1 slice of cheese. Top with remaining bread slices. Cut each sandwich in half.